DOCSLIB.ORG

Unbounded Spigot Algorithms for the Digits of Pi

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Unbounded Spigot Algorithms for the Digits of Pi Unbounded Spigot Algorithms for the Digits of Pi Jeremy Gibbons 1 INTRODUCTION. Rabinowitz and Wagon [8] present a “remarkable” algorithm for com- puting the decimal digits of π, based on the expansion ∞ (i!)22i+1 π = ∑ . (1) i=0 (2i + 1)! Their algorithm uses only bounded integer arithmetic, and is surprisingly efficient. Moreover, it admits extremely concise implementations. Witness, for example, the following (deliberately obfuscated) C pro- gram due to Dik Winter and Achim Flammenkamp [1, p. 37], which produces the first 15,000 decimal digits of π: a[52514],b,c=52514,d,e,f=1e4,g,h;main(){for(;b=c-=14;h=printf("%04d", e+d/f))for(e=d%=f;g=--b*2;d/=g)d=d*b+f*(h?a[b]:f/5),a[b]=d%--g;} Rabinowitz and Wagon call their algorithm a spigot algorithm, because it yields digits incrementally and does not reuse digits after they have been computed. The digits drip out one by one, as if from a leaky tap. In contrast, most algorithms for computing the digits of π execute inscrutably, delivering no output until the whole computation is completed. However, the Rabinowitz–Wagon algorithm has its weaknesses. In particular, the computation is in- herently bounded: one has to commit in advance to computing a certain number of digits. Based on this commitment, the computation proceeds on an appropriate finite prefix of the infinite series (1). In fact, it is essentially impossible to determine in advance how big that finite prefix should be for a given number of digits—specifically, a computation that terminates with nines for the last few digits of the output is inconclusive, because there may be a “carry” from the first few truncated terms. Rabinowitz and Wagon suggest that “in practice, one might ask for, say, six extra digits, reducing the odds of this problem to one in a million” [8, p. 197], a not entirely satisfactory recommendation. Indeed, the implementation printed at the end of their paper is not quite right [1, p. 82], sometimes printing an incorrect last digit because the finite approximation of the infinite series is one term too short. We propose a different algorithm, based on the same series (1) for π but avoiding these problems. We also show the same technique applied to other characterizations of π. No commitment need be made in advance to the number of digits to be computed; given enough memory, the programs will generate digits ad infinitum. Once more (necessarily, in fact, given the previous property), the programs are spigot algorithms in Rabinowitz and Wagon’s sense: they yield digits incrementally and do not reuse them after producing them. Of course, no algorithm using a bounded amount of memory can generate a nonrepeat- ing sequence such as the digits of π indefinitely, so we have to allow arbitrary-precision arithmetic, or some other manifestation of dynamic memory allocation. Like Rabinowitz and Wagon’s algorithm, our proposals are not competitive with state-of-the-art arithmetic-geometric mean algorithms for computing π Copyright the Mathematical Association of America 2005. All rights reserved. 1 [2], [9]. Nevertheless, our algorithms are simple to understand and admit almost as concise an implemen- tation. As evidence to support the second claim, here is a (deliberately obscure) program that will generate as many digits of π as memory will allow: > pi = g(1,0,1,1,3,3) where > g(q,r,t,k,n,l) = if 4*q+r-t<n*t > then n : g(10*q,10*(r-n*t),t,k,div(10*(3*q+r))t-10*n,l) > else g(q*k,(2*q+r)*l,t*l,k+1,div(q*(7*k+2)+r*l)(t*l),l+2) The remainder of this paper provides a justification for the foregoing program, and some others like it. These algorithms exhibit a pattern that we call streaming [3]. Informally, a streaming algorithm con- sumes a (potentially infinite) sequence of inputs and generates a (possibly infinite) sequence of outputs, maintaining some state as it goes. Based on the current state, at each step there is a choice between producing an element of the output and consuming an element of the input. Streaming seems to be a com- mon pattern for various kinds of representation changers, including several data compression and number conversion algorithms. The program under discussion is written in Haskell [5], a lazy functional programming language. As a secondary point of this paper, we hope to convince the reader that such languages are excellent vehicles for expressing mathematical computations, certainly when compared with other general-purpose programming languages such as Java, C, and Pascal, and arguably even when compared with computer algebra systems such as Mathematica. In particular, a lazy language allows direct computations with infinite data structures, which require some kind of indirect representation in most other languages. The Haskell program presented earlier has been compressed to compete with the C program for conciseness, so we do not argue that this particular one is easy to follow—but we do claim that the later Haskell programs are. 2 LAZY FUNCTIONAL PROGRAMMING IN HASKELL. To aid the reader’s understanding, we start with a brief (and necessarily incomplete) description of the concepts of functional programming (henceforth FP) and laziness and their manifestation in Haskell [5], the de facto standard lazy FP language. Further resources, including pointers to tutorials and free implementations for many platforms, can be found at the Haskell website [6]. FP is programming with expressions rather than statements. Everything is a value, and there are no assignments or other state-changing commands. Therefore, a pure FP language is referentially transparent: an expression may always be substituted for one with an equal value, without changing the meaning of the surrounding context. This makes reasoning in FP languages just like reasoning in high-school algebra. Here is a simple Haskell program: > square :: Integer -> Integer > square x = x * x Program text is marked with a “>” in the left-hand column, and comments are unmarked. This is a simple form of literate programming, in which the emphasis is placed on making the program easy for people rather than computers to read. Code and documentation are freely interspersed; indeed, the manuscript for this article is simultaneously an executable Haskell program. The first line in the square program is a type declaration; the symbol “::” should be read “has type.” Thus, square has type Integer -> Integer, that is, it is a function from Integers to Integers. (Type declarations in Haskell are nearly always optional, because they can be inferred, but we often specify them anyway for clarity.) The second line gives a definition, as an equation: in any context, a subexpression of the form square x for any x may be replaced safely by x * x. 2 Lists are central to FP. Haskell uses square brackets for lists, so [1,2,3] has three elements, and [] is the empty list. The operator “:” prefixes an element; so [1,2,3] = 1 : (2 : (3 : [])). For any type a, there is a corresponding type [a] of lists with elements drawn from type a, so, for exam- ple, [1,2,3] :: [Integer]. Haskell has a very convenient list comprehension notation, analogous to set comprehensions; for instance, [ square x | x <- [1,2,3] ] denotes the list of squares [1,4,9]. For the “substitution of equals for equals” property to be universally valid, it is important not to evaluate expressions unless their values are needed. For example, consider the following Haskell program: > three :: Integer -> Integer > three x = 3 > nonsense :: Integer > nonsense = 1 + nonsense The first definition is of a function that ignores its argument x and always returns the integer 3; the second is of a value of type Integer, but one whose evaluation never terminates. For substitutivity to hold, and in particular for three nonsense to evaluate to 3 as the equation suggests, it is important not to evaluate the function argument nonsense in the function application three nonsense. With lazy evaluation, in which evaluation is demand-driven, no expression is evaluated unless and until its value is needed to make further progress. A useful by-product of lazy evaluation is the ability to handle infinite data structures: they are evaluated only as far as is necessary. We illustrate this with a definition of the infinite sequence of Fibonacci numbers: > fibs :: [Integer] > fibs = f (0,1) where f (a,b) = a : f (b,a+b) (Here, (0,1) is a pair of Integers, and the where clause introduces a local definition.) Evaluating fibs never terminates, of course, but computing a finite prefix of it does. For example, with > take :: Integer -> [Integer] -> [Integer] > take 0 xs = [] > take (n+1) [] = [] > take (n+1) (x:xs) = x : take n xs (so take takes two arguments, an Integer and a [Integer], and returns another [Integer]), we have take 10 fibs = [0,1,1,2,3,5,8,13,21,34], which terminates normally. Functions may be polymorphic, defined for arbitrary types. Thus, the function take in fact works for lists of any element type, since elements are merely copied and not further analyzed. The most general type assignable to take is Integer -> [a] -> [a] for an arbitrary type a. Because this is FP, functions are “first-class citizens” of the language, with all the rights of any other type. Among other things, they may be passed as arguments to and returned as results from higher-order functions. For example, with the definition > map :: (a->b) -> [a] -> [b] > map f [] = [] > map f (x:xs) = f x : map f xs the list comprehension [ square x | x <- [1,2,3] ] is equal to map square [1,2,3].
Load more

Recommended publications
  • “Scrap Your Boilerplate” Reloaded
    “Scrap Your Boilerplate” Reloaded Ralf Hinze1, Andres L¨oh1, and Bruno C. d. S. Oliveira2 1 Institut f¨urInformatik III, Universit¨atBonn R¨omerstraße164, 53117 Bonn, Germany {ralf,loeh}@informatik.uni-bonn.de 2 Oxford University Computing Laboratory Wolfson Building, Parks Road, Oxford OX1 3QD, UK [email protected] Abstract. The paper “Scrap your boilerplate” (SYB) introduces a com- binator library for generic programming that offers generic traversals and queries. Classically, support for generic programming consists of two es- sential ingredients: a way to write (type-)overloaded functions, and in- dependently, a way to access the structure of data types. SYB seems to lack the second. As a consequence, it is difficult to compare with other approaches such as PolyP or Generic Haskell. In this paper we reveal the structural view that SYB builds upon. This allows us to define the combinators as generic functions in the classical sense. We explain the SYB approach in this changed setting from ground up, and use the un- derstanding gained to relate it to other generic programming approaches. Furthermore, we show that the SYB view is applicable to a very large class of data types, including generalized algebraic data types. 1 Introduction The paper “Scrap your boilerplate” (SYB) [1] introduces a combinator library for generic programming that offers generic traversals and queries. Classically, support for generic programming consists of two essential ingredients: a way to write (type-)overloaded functions, and independently, a way to access the structure of data types. SYB seems to lacks the second, because it is entirely based on combinators.
    [Show full text]
  • Manydsl One Host for All Language Needs
    ManyDSL One Host for All Language Needs Piotr Danilewski March 2017 A dissertation submitted towards the degree (Dr.-Ing.) of the Faculty of Mathematics and Computer Science of Saarland University. Saarbrücken Dean Prof. Dr. Frank-Olaf Schreyer Date of Colloquium June 6, 2017 Examination Board: Chairman Prof. Dr. Sebastian Hack Reviewers Prof. Dr.-Ing. Philipp Slusallek Prof. Dr. Wilhelm Reinhard Scientific Asistant Dr. Tim Dahmen Piotr Danilewski, [email protected] Saarbrücken, June 6, 2017 Statement I hereby declare that this dissertation is my own original work except where otherwise indicated. All data or concepts drawn directly or indirectly from other sources have been correctly acknowledged. This dissertation has not been submitted in its present or similar form to any other academic institution either in Germany or abroad for the award of any degree. Saarbrücken, June 6, 2017 (Piotr Danilewski) Declaration of Consent Herewith I agree that my thesis will be made available through the library of the Computer Science Department. Saarbrücken, June 6, 2017 (Piotr Danilewski) Zusammenfassung Die Sprachen prägen die Denkweise. Das ist die Tatsache für die gesprochenen Sprachen aber auch für die Programmiersprachen. Da die Computer immer wichtiger in jedem Aspekt des menschlichen Lebens sind, steigt der Bedarf um entsprechend neue Konzepte in den Programmiersprachen auszudrücken. Jedoch, damit unsere Denkweise sich weiterentwicklen könnte, müssen sich auch die Programmiersprachen weiterentwickeln. Aber welche Hilfsmittel gibt es um die Programmiersprachen zu schaffen und aufzurüsten? Wie kann man Entwickler ermutigen damit sie eigene Sprachen definieren, die dem Bereich in dem sie arbeiten am besten passen? Heutzutage gibt es zwei Methoden.
    [Show full text]
  • On Specifying and Visualising Long-Running Empirical Studies
    On Specifying and Visualising Long-Running Empirical Studies Peter Y. H. Wong and Jeremy Gibbons Computing Laboratory, University of Oxford, United Kingdom fpeter.wong,[email protected] Abstract. We describe a graphical approach to formally specifying tem- porally ordered activity routines designed for calendar scheduling. We introduce a workflow model OWorkflow, for constructing specifications of long running empirical studies such as clinical trials in which obser- vations for gathering data are performed at strict specific times. These observations, either manually performed or automated, are often inter- leaved with scientific procedures, and their descriptions are recorded in a calendar for scheduling and monitoring to ensure each observation is carried out correctly at a specific time. We also describe a bidirectional transformation between OWorkflow and a subset of Business Process Modelling Notation (BPMN), by which graphical specification, simula- tion, automation and formalisation are made possible. 1 Introduction A typical long-running empirical study consists of a series of scientific proce- dures interleaved with a set of observations performed over a period of time; these observations may be manually performed or automated, and are usually recorded in a calendar schedule. An example of a long-running empirical study is a clinical trial, where observations, specifically case report form submissions, are performed at specific points in the trial. In such examples, observations are interleaved with clinical interventions on patients; precise descriptions of these observations are then recorded in a patient study calendar similar to the one shown in Figure 1(a). Currently study planners such as trial designers supply information about observations either textually or by inputting textual infor- mation and selecting options on XML-based data entry forms [2], similar to the one shown in Figure 1(b).
    [Show full text]
  • Lecture Notes in Computer Science 6120 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan Van Leeuwen
    Lecture Notes in Computer Science 6120 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany Claude Bolduc Jules Desharnais Béchir Ktari (Eds.) Mathematics of Program Construction 10th International Conference, MPC 2010 Québec City, Canada, June 21-23, 2010 Proceedings 13 Volume Editors Claude Bolduc Jules Desharnais Béchir Ktari Université Laval, Département d’informatique et de génie logiciel Pavillon Adrien-Pouliot, 1065 Avenue de la Médecine Québec, QC, G1V 0A6, Canada E-mail: {Claude.Bolduc, Jules.Desharnais, Bechir.Ktari}@ift.ulaval.ca Library of Congress Control Number: 2010927075 CR Subject Classification (1998): F.3, D.2, F.4.1, D.3, D.2.4, D.1 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN 0302-9743 ISBN-10 3-642-13320-7 Springer Berlin Heidelberg New York ISBN-13 978-3-642-13320-6 Springer Berlin Heidelberg New York This work is subject to copyright.
    [Show full text]
  • Domain-Specific Languages for Modeling and Simulation
    Domain-specifc Languages for Modeling and Simulation Dissertation zur Erlangung des akademischen Grades Doktor-Ingenieur (Dr.-Ing.) der Fakultät für Informatik und Elektrotechnik der Universität Rostock vorgelegt von Tom Warnke, geb. am 25.01.1988 in Malchin aus Rostock Rostock, 14. September 2020 https://doi.org/10.18453/rosdok_id00002966 Dieses Werk ist lizenziert unter einer Creative Commons Namensnennung - Weitergabe unter gleichen Bedingungen 4.0 International Lizenz. Gutachter: Prof. Dr. Adelinde M. Uhrmacher (Universität Rostock) Prof. Rocco De Nicola (IMT Lucca) Prof. Hans Vangheluwe (Universität Antwerpen) Eingereicht am 14. September 2020 Verteidigt am 8. Januar 2021 Abstract Simulation models and simulation experiments are increasingly complex. One way to handle this complexity is developing software languages tailored to specifc application domains, so-called domain-specifc languages (DSLs). This thesis explores the potential of employing DSLs in modeling and simulation. We study diferent DSL design and implementation techniques and illustrate their benefts for expressing simulation models as well as simulation experiments with several examples. Regarding simulation models, we focus on discrete-event models based on continuous- time Markov chains (CTMCs). Most of our work revolves around ML-Rules, an rule-based modeling language for biochemical reaction networks. First, we relate the expressive power of ML-Rules to other currently available modeling languages for this application domain. Then we defne the abstract syntax and operational semantics for ML-Rules, mapping models to CTMCs in an unambiguous and precise way. Based on the formal defnitions, we present two approaches to implement ML-Rules as a DSL. The core of both implementations is fnding the matches for the patterns on the left side of ML-Rules’ rules.
    [Show full text]
  • Towards a Repository of Bx Examples
    Towards a Repository of Bx Examples James Cheney, James McKinna, Jeremy Gibbons Perdita Stevens Department of Computer Science School of Informatics University of Oxford University of Edinburgh fi[email protected] fi[email protected] ABSTRACT that researchers in bx will both add their examples to the reposi- We argue for the creation of a curated repository of examples of tory, and refer to examples from the repository as appropriate. This bidirectional transformations (bx). In particular, such a resource should have benefits both for authors and for readers. Naturally, may support research on bx, especially cross-fertilisation between papers will still have to be sufficiently self-contained; but we think the different communities involved. We have initiated a bx reposi- that the repository may still be helpful. For example, if one’s paper tory, which is introduced in this paper. We discuss our design deci- illustrates some new work using an existing example, one might sions and their rationale, and illustrate them using the now classic describe the example briefly (as at present), focusing on the as- Composers example. We discuss the difficulties that this undertak- pects most relevant to the work at hand. One can, however, also ing may face, and comment on how they may be overcome. give a reference into the repository, where there is more detail and discussion of the example, which some readers may find helpful. Over time, we might expect that certain examples in the repository 1. INTRODUCTION become familiar to most researchers in bx. Then, if a paper says Research into bidirectional transformations (henceforth: bx) in- it uses such an example, the reader’s task is eased.
    [Show full text]
  • A Separation Logic for Concurrent Randomized Programs
    A Separation Logic for Concurrent Randomized Programs JOSEPH TASSAROTTI, Carnegie Mellon University, USA ROBERT HARPER, Carnegie Mellon University, USA We present Polaris, a concurrent separation logic with support for probabilistic reasoning. As part of our logic, we extend the idea of coupling, which underlies recent work on probabilistic relational logics, to the setting of programs with both probabilistic and non-deterministic choice. To demonstrate Polaris, we verify a variant of a randomized concurrent counter algorithm and a two-level concurrent skip list. All of our results have been mechanized in Coq. CCS Concepts: • Theory of computation → Separation logic; Program verification; Additional Key Words and Phrases: separation logic, concurrency, probability ACM Reference Format: Joseph Tassarotti and Robert Harper. 2019. A Separation Logic for Concurrent Randomized Programs. Proc. ACM Program. Lang. 3, POPL, Article 64 (January 2019), 31 pages. https://doi.org/10.1145/3290377 1 INTRODUCTION Many concurrent algorithms use randomization to reduce contention and coordination between threads. Roughly speaking, these algorithms are designed so that if each thread makes a local random choice, then on average the aggregate behavior of the whole system will have some good property. For example, probabilistic skip lists [53] are known to work well in the concurrent setting [25, 32], because threads can independently insert nodes into the skip list without much synchronization. In contrast, traditional balanced tree structures are difficult to implement in a scalable way because re-balancing operations may require locking access to large parts of the tree. However, concurrent randomized algorithms are difficult to write and reason about. The useof just concurrency or randomness alone makes it hard to establish the correctness of an algorithm.
    [Show full text]
  • Profunctor Optics Modular Data Accessors
    Profunctor Optics Modular Data Accessors Matthew Pickeringa, Jeremy Gibbonsb, and Nicolas Wua a University of Bristol b University of Oxford Abstract Data accessors allow one to read and write components of a data structure, such as the fields of a record, the variants of a union, or the elements of a container. These data accessors are collectively known as optics; they are fundamental to programs that manipulate complex data. Individual data accessors for simple data structures are easy to write, for example as pairs of ‘getter’ and ‘setter’ methods. However, it is not obvious how to combine data accessors, in such a way that data accessors for a compound data structure are composed out of smaller data accessors for the parts of that structure. Generally, one has to write a sequence of statements or declarations that navigate step by step through the data structure, accessing one level at a time—which is to say, data accessors are traditionally not first-class citizens, combinable in their own right. We present a framework for modular data access, in which individual data accessors for simple data struc- tures may be freely combined to obtain more complex data accessors for compound data structures. Data accessors become first-class citizens. The framework is based around the notion of profunctors, a flexible gen- eralization of functions. The language features required are higher-order functions (‘lambdas’ or ‘closures’), parametrized types (‘generics’ or ‘abstract types’) of higher kind, and some mechanism for separating inter- faces from implementations (‘abstract classes’ or ‘modules’). We use Haskell as a vehicle in which to present our constructions, but other languages such as Scala that provide the necessary features should work just as well.
    [Show full text]
  • A Generative Approach to Traversal-Based Generic Programming
    A Generative Approach to Traversal-based Generic Programming Bryan Chadwick Karl Lieberherr College of Computer & Information Science Northeastern University, 360 Huntington Avenue Boston, Massachusetts 02115 USA. {chadwick, lieber}@ccs.neu.edu Abstract • We introduce a new form of traversal-based generic program- ming that uses function objects to fold over structures (Sec- The development of complex software requires the implementa- tion 4). Functions are flexible, extensible, and written indepen- tion of functions over a variety of recursively defined data struc- dent of the traversal implementation using a variation of multi- tures. Much of the corresponding code is not necessarily diffi- ple dispatch. This provides a special form of shape polymor- cult, but more tedious and/or repetitive and sometimes easy to get phism with support for both general and specialized generic wrong. Data structure traversals fall into this category, particu- functions (Section 5). larly in object-oriented languages where traversal code is spread throughout many cooperating modules. In this paper we present a • Our approach is supported by a class generator that consumes new form of generic programming using traversals that lends it- a concise description of data types and produces Java classes self to a flexible, safe, and efficient generative implementation. We along with specific instances of generic functions like parse(), describe the approach, its relation to generic and generative pro- print(), and equals() (Section 6). The generative frame- gramming, and our implementation and resulting performance. work is extensible, so programmers can add their own generic functions parametrized by datatype definitions. • Function objects (specific and generic) can be type checked 1.
    [Show full text]
  • Denotational Translation Validation
    Denotational Translation Validation The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Govereau, Paul. 2012. Denotational Translation Validation. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:10121982 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA c 2012 - Paul Govereau All rights reserved. Thesis advisor Author Greg Morrisett Paul Govereau Denotational Translation Validation Abstract In this dissertation we present a simple and scalable system for validating the correctness of low-level program transformations. Proving that program transfor- mations are correct is crucial to the development of security critical software tools. We achieve a simple and scalable design by compiling sequential low-level programs to synchronous data-flow programs. Theses data-flow programs are a denotation of the original programs, representing all of the relevant aspects of the program se- mantics. We then check that the two denotations are equivalent, which implies that the program transformation is semantics preserving. Our denotations are computed by means of symbolic analysis. In order to achieve our design, we have extended symbolic analysis to arbitrary control-flow graphs. To this end, we have designed an intermediate language called Synchronous Value Graphs (SVG), which is capable of representing our denotations for arbitrary control-flow graphs, we have built an algorithm for computing SVG from normal assembly language, and we have given a formal model of SVG which allows us to simplify and compare denotations.
    [Show full text]
  • Program Design by Calculation
    J.N. OLIVEIRA University of Minho PROGRAM DESIGN BY CALCULATION (DRAFT of textbook in preparation) Last update: July 2021 CONTENTS Preamble 1 1 INTRODUCTION 3 1.1 A bit of archaeology 3 1.2 The future ahead 9 1.3 Summary 12 1.4 Why this Book? 14 ICALCULATINGWITHFUNCTIONS 15 2 ANINTRODUCTIONTOPOINTFREEPROGRAMMING 16 2.1 Introducing functions and types 17 2.2 Functional application 18 2.3 Functional equality and composition 18 2.4 Identity functions 21 2.5 Constant functions 21 2.6 Monics and epics 23 2.7 Isos 24 2.8 Gluing functions which do not compose — products 25 2.9 Gluing functions which do not compose — coproducts 31 2.10 Mixing products and coproducts 34 2.11 Elementary datatypes 36 2.12 Natural properties 39 2.13 Universal properties 40 2.14 Guards and McCarthy’s conditional 43 2.15 Gluing functions which do not compose — exponen- tials 46 2.16 Finitary products and coproducts 53 2.17 Initial and terminal datatypes 54 2.18 Sums and products in HASKELL 56 2.19 Exercises 59 2.20 Bibliography notes 62 3 RECURSIONINTHEPOINTFREESTYLE 64 3.1 Motivation 64 3.2 From natural numbers to finite sequences 69 3.3 Introducing inductive datatypes 74 3.4 Observing an inductive datatype 79 3.5 Synthesizing an inductive datatype 82 3.6 Introducing (list) catas, anas and hylos 84 3.7 Inductive types more generally 88 3.8 Functors 89 3.9 Polynomial functors 91 3.10 Polynomial inductive types 93 3.11 F-algebras and F-homomorphisms 94 ii Contents iii 3.12 F-catamorphisms 94 3.13 Parameterization and type functors 97 3.14 A catalogue of standard polynomial
    [Show full text]
  • (Pdf) of the School of Squiggol: a History of the Bird−Meertens
    The School of Squiggol A History of the Bird{Meertens Formalism Jeremy Gibbons University of Oxford Abstract. The Bird{Meertens Formalism, colloquially known as \Squig- gol", is a calculus for program transformation by equational reasoning in a function style, developed by Richard Bird and Lambert Meertens and other members of IFIP Working Group 2.1 for about two decades from the mid 1970s. One particular characteristic of the development of the Formalism is fluctuating emphasis on novel `squiggly' notation: sometimes favouring notational exploration in the quest for conciseness and precision, and sometimes reverting to simpler and more rigid nota- tional conventions in the interests of accessibility. This paper explores that historical ebb and flow. 1 Introduction In 1962, IFIP formed Working Group 2.1 to design a successor to the seminal algorithmic language Algol 60 [4]. WG2.1 eventually produced the specification for Algol 68 [63, 64]|a sophisticated language, presented using an elaborate two- level description notation, which received a mixed reception. WG2.1 continues to this day; technically, it retains responsibility for the Algol languages, but practi- cally it takes on a broader remit under the current name Algorithmic Languages and Calculi. Over the years, the Group has been through periods of focus and periods of diversity. But after the Algol 68 project, the period of sharpest focus covered the two decades from the mid 1970s to the early 1990s, when what later became known as the Bird{Meertens Formalism (BMF) drew the whole group together again. It is the story of those years that is the subject of this paper.
    [Show full text]